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Oscillators of a Spring
Storyboard 
In the case of the spring the force is proportional to the elongation of the spring so that the equations of motion are linear and the frequency of the oscillation is independent of the amplitude. This is the key to generate an oscillation that does not depend on the fact that the friction decreases over time. This is why old clocks used (circular) springs to generate stable oscillations to measure elapsed time.
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Oscillations with a spring
Description
One of the systems it depicts is that of a spring. This is associated with the elastic deformation of the material from which the spring is made. By "elastic," we mean a deformation that, upon removing the applied stress, allows the system to fully regain its original shape. It's understood that it doesn't undergo plastic deformation.
Since the energy of the spring is given by
$E=\displaystyle\frac{1}{2}m_i v^2+\displaystyle\frac{1}{2}k x^2$
the period will be equal to
$T=2\pi\sqrt{\displaystyle\frac{m_i}{k}}$
and thus, the angular frequency is
| $ \omega_0 ^2=\displaystyle\frac{ k }{ m_i }$ |
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Model
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Parameters
Variables
Calculations
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Calculations
Calculations
Variable 
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Target : Equation To be used 
Equations
$ E = K + V $
E = K + V
$ K =\displaystyle\frac{ p ^2}{2 m_i }$
K = p ^2/(2 * m_i )
$ \nu =\displaystyle\frac{1}{ T }$
nu =1/ T
$ \omega = 2 \pi \nu $
omega = 2* pi * nu
$ \omega = \displaystyle\frac{2 \pi }{ T }$
omega = 2* pi / T
$ \omega_0 ^2=\displaystyle\frac{ k }{ m_i }$
omega_0 ^2 = k / m_i
$ p = m_i v $
p = m_i * v
$ T =2 \pi \sqrt{\displaystyle\frac{ m_i }{ k }}$
T =2* pi *sqrt( m_i / k )
$ v = - x_0 \omega_0 \sin \omega_0 t $
v = - x_0 * omega_0 *sin( omega_0 * t )
$ V =\displaystyle\frac{1}{2} k x ^2$
V = k * x ^2/2
$ E =\displaystyle\frac{1}{2} k x_0 ^2$
V = k * x ^2/2
$ x = x_0 \cos \omega t $
x = x_0 *cos( omega_0 * t )
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Total Energy
Equation
The total energy corresponds to the sum of the total kinetic energy and the potential energy:
| $ E = K + V $ |
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Kinetic energy as a function of moment
Equation
The kinetic energy of a mass $m$
| $ K_t =\displaystyle\frac{1}{2} m_i v ^2$ |
can be expressed in terms of momentum as
| $ K =\displaystyle\frac{ p ^2}{2 m_i }$ |
Since kinetic energy is equal to
| $ K_t =\displaystyle\frac{1}{2} m_i v ^2$ |
and momentum is
| $ p = m_i v $ |
we can express it as
$K_t=\displaystyle\frac{1}{2} m_i v^2=\displaystyle\frac{1}{2} m_i \left(\displaystyle\frac{p}{m_i}\right)^2=\displaystyle\frac{p^2}{2m_i}$
or
| $ K =\displaystyle\frac{ p ^2}{2 m_i }$ |
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Elastic potential energy (1)
Equation
En el caso elástico (resorte) la fuerza es
la energía
| $ dW = \vec{F} \cdot d\vec{s} $ |
se puede mostrar que en este caso es
| $ V =\displaystyle\frac{1}{2} k x ^2$ |
En el caso elástico (resorte) la fuerza es
con
| $ dW = \vec{F} \cdot d\vec{s} $ |
\\n\\nLa diferencia\\n\\n
$\Delta x = x_2 - x_1$
\\n\\ncorresponde al camino recorrido por lo que\\n\\n
$\Delta W=k,x,\Delta x=k(x_2-x_1)\displaystyle\frac{(x_1+x_2)}{2}=\displaystyle\frac{k}{2}(x_2^2-x_1^2)$
y con ello la energía potencial elástica es
| $ V =\displaystyle\frac{1}{2} k x ^2$ |
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Elastic potential energy (2)
Equation
En el caso elástico (resorte) la fuerza es
la energía
| $ dW = \vec{F} \cdot d\vec{s} $ |
se puede mostrar que en este caso es
| $ E =\displaystyle\frac{1}{2} k x_0 ^2$ |
| $ V =\displaystyle\frac{1}{2} k x ^2$ |
En el caso elástico (resorte) la fuerza es
con
| $ dW = \vec{F} \cdot d\vec{s} $ |
\\n\\nLa diferencia\\n\\n
$\Delta x = x_2 - x_1$
\\n\\ncorresponde al camino recorrido por lo que\\n\\n
$\Delta W=k,x,\Delta x=k(x_2-x_1)\displaystyle\frac{(x_1+x_2)}{2}=\displaystyle\frac{k}{2}(x_2^2-x_1^2)$
y con ello la energía potencial elástica es
| $ V =\displaystyle\frac{1}{2} k x ^2$ |
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Oscillations with a spring
Equation
The product of the hooke Constant ($k$) and the inertial Mass ($m_i$) is called the frecuencia angular del resorte ($\omega$) and is defined as:
| $ \omega_0 ^2=\displaystyle\frac{ k }{ m_i }$ |
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Periodo de la Oscilación
Equation
Como la oscilación cumple las leyes físicas se puede hacer uso del hecho que el area debajo de la curva velocidad vs tiempo el camino recorrido para determinar el perido. Como la velocidad es\\n\\n
$\displaystyle\int_0^{T/2}v(t)dt=\sqrt{\displaystyle\frac{2E}{m}}\displaystyle\int_0^{T/2}\cos \displaystyle\frac{2\pi t}{T}dt=\sqrt{\displaystyle\frac{2E}{m}}\displaystyle\frac{T}{\pi}$
\\n\\ny el camino entre un mínimo a un máximo de una elongación, lo que ocurre entre el tiempo
$x_{max}-x_{min}=2\sqrt{\displaystyle\frac{2E}{k}}$
se tiene que
| $ T =2 \pi \sqrt{\displaystyle\frac{ m_i }{ k }}$ |
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Frequency
Equation
The frequency ($\nu$) corresponds to the number of times an oscillation occurs within one second. The period ($T$) represents the time it takes for one oscillation to occur. Therefore, the number of oscillations per second is:
| $ \nu =\displaystyle\frac{1}{ T }$ |
Frequency is indicated in Hertz (Hz).
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Angular frequency
Equation
The angular frequency ($\omega$) is with the period ($T$) equal to
| $ \omega = \displaystyle\frac{2 \pi }{ T }$ |
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Relación frecuencia angular - frecuencia
Equation
Como la frecuencia angular es con angular frequency $rad/s$, period $s$ and pi $rad$ igual a
| $ \omega = \displaystyle\frac{2 \pi }{ T }$ |
y la frecuencia con frequency $Hz$ and period $s$ igual a
| $ \nu =\displaystyle\frac{1}{ T }$ |
se tiene que con frequency $Hz$ and period $s$ igual a
| $ \omega = 2 \pi \nu $ |
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Oscillation amplitude
Equation
With the description of the oscillation using
| $ z = x_0 \cos \omega_0 t + i x_0 \sin \omega_0 t $ |
the real part corresponds to the temporal evolution of the amplitude
| $ x = x_0 \cos \omega t $ |
| $ x = x_0 \cos \omega_0 t $ |
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Swing speed
Equation
When we extract the real part of the derivative of the complex number representing the oscillation
| $ \dot{z} = i \omega_0 z $ |
whose real part corresponds to the velocity
| $ v = - x_0 \omega \sin \omega t $ |
| $ v = - x_0 \omega_0 \sin \omega_0 t $ |
Using the complex number
| $ z = x_0 \cos \omega_0 t + i x_0 \sin \omega_0 t $ |
introduced in
| $ \dot{z} = i \omega_0 z $ |
we obtain
$\dot{z} = i\omega_0 z = i \omega_0 x_0 \cos \omega_0 t - \omega_0 x_0 \sin \omega_0 t$
thus, the velocity is obtained as the real part
| $ v = - x_0 \omega_0 \sin \omega_0 t $ |
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